Group #30774388 — subgroups

Subgroups — #30774388

Order 17152.

366 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863794	C₂	1	1	yes	maximal, characteristic
2	#22560088	C₂	1	1	yes	maximal, characteristic
3	#22558315	C₂	1	2	yes	maximal
4	#22558308	C₂	1	2	yes	maximal
5	C₂×#364×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14524043	—	2	1	no	—
7	#16863794	—	2	2	no	—
8	#16863794	C₄	1	4	yes	complemented
9	C₂×#14523979	C₂²	1	1	yes	characteristic
10	#16863434	C₂²	1	1	yes	characteristic
11	C₂×#14523975	—	2	1	no	—
12	C₂×#14523975	C₂²	1	2	yes	—
13	#16863351	—	2	1	no	—
14	#16863156	—	2	1	no	—
15	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	#364×C₆₇	—	2	2	no	—
18	C₂²×Dic₁₃₄	—	4	1	no	—
19	C₂×#11761612	—	2	1	no	—
20	#14524043	—	2	4	no	—
21	#14524043	—	4	2	no	—
22	#14524002	—	2	1	no	—
23	C₂×C₄⋊Dic₆₇	—	2	2	no	—
24	C₂×C₄⋊Dic₆₇	—	4	1	no	—
25	#14523979	—	2	1	no	—
26	#14523979	Q₈	1	2	yes	complemented
27	#14523977	Q₈	1	2	yes	complemented
28	#14523975	—	2	4	no	—
29	#14523975	—	2	2	no	—
30	#14523975	—	2	1	no	—
31	#14523975	C₂×C₄	1	4	yes	—
32	#14523953	—	4	1	no	—
33	C₈×Dic₆₇	—	4	1	no	—
34	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
37	M₃₂×C₆₇	—	2	2	no	—
38	M₃₂×C₆₇	—	2	2	no	—
39	C₂×C₁₆×C₆₇	—	2	2	no	—
40	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₆₇	D₄	1	2	yes	—
42	C₂×Dic₁₃₄	—	4	4	no	—
43	C₂×Dic₁₃₄	—	8	4	no	—
44	C₂²×Dic₆₇	—	4	2	no	—
45	#11761612	—	2	2	no	—
46	#11761612	—	4	1	no	—
47	C₂×C₆₇⋊C₈	—	4	1	no	—
48	C₄⋊Dic₆₇	—	2	4	no	—
49	C₄⋊Dic₆₇	—	2	2	no	—
50	C₄⋊Dic₆₇	—	4	4	no	—
51	C₄⋊Dic₆₇	—	4	2	no	—
52	C₂²⋊Dic₆₇	—	4	1	no	—
53	C₄×Dic₆₇	—	4	1	no	—
54	C₂²×C₄×C₆₇	—	2	1	no	—
55	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
57	C₂×C₈×C₆₇	SD₁₆	1	1	yes	characteristic
58	C₂×C₈×C₆₇	SD₁₆	1	1	yes	characteristic
59	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
60	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
65	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
66	C₄²×C₆₇	—	2	1	no	—
67	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
68	C₁₆×C₆₇	—	4	2	no	—
69	C₂³×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
70	C₂×C₄×C₆₇	—	2	4	no	—
71	C₂×C₄×C₆₇	—	2	2	no	—
72	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
77	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₂×Dic₆₇	—	4	4	no	—
79	C₂×Dic₆₇	—	4	2	no	—
80	C₂×Dic₆₇	—	8	4	no	—
81	Dic₁₃₄	—	8	8	no	—
82	C₈×C₆₇	#103	1	2	yes	—
83	C₈×C₆₇	#103	1	2	yes	—
84	C₈×C₆₇	C₄×Q₈	1	1	yes	characteristic
85	C₈×C₆₇	C₄×Q₈	1	1	yes	characteristic
86	C₈×C₆₇	#124	1	1	yes	characteristic
87	C₈×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
88	C₆₇⋊C₈	—	4	2	no	—
89	C₂²×C₆₇	—	2	2	no	—
90	C₂²×C₆₇	#418	1	1	yes	characteristic
91	C₂²×C₆₇	#476	1	1	yes	characteristic
92	C₂²×C₆₇	#477	1	1	yes	characteristic
93	C₄×C₆₇	—	2	4	no	—
94	C₄×C₆₇	#391	1	1	yes	characteristic
95	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
96	C₄×C₆₇	#476	1	1	yes	characteristic
97	C₄×C₆₇	#477	1	1	yes	characteristic
98	Dic₆₇	—	8	4	no	—
99	Dic₆₇	—	8	2	no	—
100	#13272	—	67	1	no	maximal, Sylow
101	C₂×C₆₇	—	2	1	no	—
102	C₂×C₆₇	—	2	1	no	—
103	C₂×C₆₇	#1987	1	1	yes	characteristic
104	C₂×C₆₇	#2230	1	2	yes	—
105	C₂×#498	—	67	1	no	—
106	C₂×#364	D₆₇	1	1	yes	characteristic
107	#1772	—	67	1	no	—
108	#1386	—	67	2	no	—
109	#1383	—	67	2	no	—
110	C₆₇	#13272	1	1	yes	characteristic, complemented, Sylow
111	C₂×C₄⋊Q₈	—	134	1	no	—
112	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
113	#498	—	67	4	no	—
114	#498	—	134	2	no	—
115	C₂×#108	—	67	1	no	—
116	#427	—	67	1	no	—
117	C₂×#106	—	67	2	no	—
118	C₂×#106	—	134	1	no	—
119	#405	—	134	1	no	—
120	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
121	#364	—	2	2	no	—
122	#339	—	134	1	no	—
123	C₂²×Q₈	—	268	1	no	—
124	C₂×M₁₆	—	134	1	no	—
125	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
127	C₄⋊Q₈	—	268	2	no	—
128	C₄⋊Q₈	—	134	4	no	—
129	#117	—	134	1	no	—
130	C₂×C₄⋊C₄	—	268	1	no	—
131	C₂×C₄⋊C₄	—	134	2	no	—
132	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
133	M₃₂	—	2	2	no	—
134	M₃₂	—	2	2	no	—
135	C₂×C₁₆	—	2	2	no	—
136	#108	—	67	2	no	—
137	#108	—	134	1	no	—
138	#107	—	67	2	no	—
139	#106	—	67	4	no	—
140	#106	—	134	4	no	—
141	#106	—	134	2	no	—
142	#106	—	134	1	no	—
143	#97	—	268	1	no	—
144	C₄×C₈	—	268	1	no	—
145	C₄×C₈	C₄×D₆₇	1	2	yes	—
146	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
147	C₂×Q₈	—	268	4	no	—
148	C₂×Q₈	—	536	4	no	—
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	—	268	2	no	—
151	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
152	M₁₆	—	268	1	no	—
153	M₁₆	—	134	2	no	—
154	C₂×C₈	—	268	1	no	—
155	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
156	C₂×C₈	#11761617	1	1	yes	characteristic
157	C₂×C₈	#11761617	1	1	yes	characteristic
158	C₂×C₈	Dic₂₆₈	1	2	yes	—
159	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
160	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
161	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
162	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
163	C₂×C₈	#11761627	1	1	yes	characteristic
164	C₂×C₈	#11761627	1	1	yes	characteristic
165	C₄⋊C₄	—	268	4	no	—
166	C₄⋊C₄	—	268	2	no	—
167	C₄⋊C₄	—	134	4	no	—
168	C₄⋊C₄	—	134	2	no	—
169	C₂²⋊C₄	—	268	1	no	—
170	C₄²	—	2	1	no	—
171	C₄²	—	268	1	no	—
172	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
173	C₁₆	—	4	2	no	—
174	C₂³	#14524071	1	1	yes	characteristic
175	Q₈	—	536	8	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	268	4	no	—
179	C₂×C₄	—	268	2	no	—
180	C₂×C₄	—	536	4	no	—
181	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
182	C₂×C₄	#14524023	1	1	yes	characteristic
183	C₂×C₄	#14524023	1	1	yes	characteristic
184	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
185	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
186	C₂×C₄	#14524071	1	1	yes	characteristic
187	C₈	—	268	2	no	—
188	C₈	#14523968	1	2	yes	—
189	C₈	#14523968	1	2	yes	—
190	C₈	#14524008	1	1	yes	characteristic
191	C₈	#14524008	1	1	yes	characteristic
192	C₈	#14524030	1	1	yes	characteristic
193	C₈	#14524041	1	1	yes	characteristic
194	C₂²	—	2	2	no	—
195	C₂²	#16863402	1	1	yes	characteristic
196	C₂²	#16863687	1	1	yes	characteristic
197	C₂²	#16863692	1	1	yes	characteristic
198	C₄	—	2	4	no	—
199	C₄	—	536	4	no	—
200	C₄	—	536	2	no	—
201	C₄	#16863301	1	1	yes	characteristic
202	C₄	C₂×#14523968	1	1	yes	characteristic
203	C₄	#16863687	1	1	yes	characteristic
204	C₄	#16863692	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	#22561156	1	1	yes	characteristic
208	C₂	#22562381	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863794

C₂

yes

maximal, characteristic

#22560088

C₂

yes

maximal, characteristic

#22558315

C₂

yes

maximal

#22558308

C₂

yes

maximal

C₂×#364×C₆₇

C₂

yes

maximal, characteristic

C₂×#14524043

—

#16863794

—

#16863794

C₄

yes

complemented

C₂×#14523979

C₂²

yes

characteristic

#16863434

C₂²

yes

characteristic

C₂×#14523975

—

C₂×#14523975

C₂²

yes

—

#16863351

—

#16863156

—

C₂×M₃₂×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#364×C₆₇

—

C₂²×Dic₁₃₄

—

—

—

—

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

#14523979

—

#14523979

Q₈

yes

complemented

#14523977

Q₈

yes

—

—

—

C₂×C₄

yes

—

#14523953

—

C₈×Dic₆₇

—

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

characteristic

M₃₂×C₆₇

—

M₃₂×C₆₇

—

C₂×C₁₆×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₂×C₈×C₆₇

SD₁₆

yes

characteristic

C₂×C₈×C₆₇

SD₁₆

yes

characteristic

C₂×C₈×C₆₇

Q₁₆

yes

—

C₂×C₈×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₇

C₂×Q₈

yes

C₂×C₈×C₆₇

yes

C₂×C₈×C₆₇

yes

C₄²×C₆₇

—

C₄²×C₆₇

C₂²⋊C₄

yes

—

C₁₆×C₆₇

—

C₂³×C₆₇

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

C₈×C₆₇

#103

yes

—

C₈×C₆₇

#103

yes

—

C₈×C₆₇

C₄×Q₈

yes

characteristic

C₈×C₆₇

C₄×Q₈

yes

characteristic

C₈×C₆₇

#124

yes

characteristic

C₈×C₆₇

C₄⋊Q₈

yes

characteristic

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

100

#13272

—

maximal, Sylow

101

C₂×C₆₇

—

102

C₂×C₆₇

—

103

C₂×C₆₇

#1987

yes

characteristic

104

C₂×C₆₇

#2230

yes

—

105

C₂×#498

—

106

C₂×#364

D₆₇

yes

characteristic

107

#1772

—

108

#1386

—

109

#1383

—

110

C₆₇

#13272

yes

characteristic, complemented, Sylow

111

C₂×C₄⋊Q₈

—

134

—

112

C₂×M₃₂

C₂×D₆₇

yes

—

113

#498

—

114

#498

—

134

—

115

C₂×#108

—

116

#427

—

117

C₂×#106

—

118

C₂×#106

—

134

—

119

#405

—

134

—

120

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

121

#364

—

122

#339

—

134

—

123

C₂²×Q₈

—

268

—

124

C₂×M₁₆

—

134

—

125

C₂²×C₈

D₂₆₈

yes

characteristic

126

C₂²×C₈

C₂²×D₆₇

yes

characteristic

127

C₄⋊Q₈

—

268

—

128

C₄⋊Q₈

—

134

—

129

#117

—

134

—

130

C₂×C₄⋊C₄

—

268

—

131

C₂×C₄⋊C₄

—

134

—

132

C₂×C₄²

D₂₆₈

yes

characteristic

133

M₃₂

—

134

M₃₂

—

135

C₂×C₁₆

—

136

#108

—

137

#108

—

134

—

138

#107

—

139

#106

—

140

#106

—

134

—

141

#106

—

134

—

142

#106

—

134

—

143

#97

—

268

—

144

C₄×C₈

—

268

—

145

C₄×C₈

C₄×D₆₇

yes

—

146

C₄×C₈

C₂²⋊D₆₇

yes

—

147

C₂×Q₈

—

268

—

148

C₂×Q₈

—

536

—

149

C₂²×C₄

—

150

C₂²×C₄

—

268

—

151

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

152

M₁₆

—

268

—

153

M₁₆

—

134

—

154

C₂×C₈

—

268

—

155

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

—

156

C₂×C₈

#11761617

yes

characteristic

157

C₂×C₈

#11761617

yes

characteristic

158

C₂×C₈

Dic₂₆₈

yes

—

159

C₂×C₈

C₂×C₄×D₆₇

yes

characteristic

160

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

161

C₂×C₈

Q₈×D₆₇

yes

characteristic

162

C₂×C₈

Q₈×D₆₇

yes

characteristic

163

C₂×C₈

#11761627

yes

characteristic

164

C₂×C₈

#11761627

yes

characteristic

165

C₄⋊C₄

—

268

—

166

C₄⋊C₄

—

268

—

167

C₄⋊C₄

—

134

—

168

C₄⋊C₄

—

134

—

169

C₂²⋊C₄

—

268

—

170

C₄²

—

171

C₄²

—

268

—

172

C₄²

(C₂×D₆₇)⋊C₄

yes

—

173

C₁₆

—

174

C₂³

#14524071

yes

characteristic

175

Q₈

—

536

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

268

—

179

C₂×C₄

—

268

—

180

C₂×C₄

—

536

—

181

C₂×C₄

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

182

C₂×C₄

#14524023

yes

characteristic

183

C₂×C₄

#14524023

yes

characteristic

184

C₂×C₄

C₂×#11761617

yes

characteristic

185

C₂×C₄

C₂×Dic₂₆₈

yes

characteristic

186

C₂×C₄

#14524071

yes

characteristic

187

C₈

—

268

—

188

C₈

#14523968

yes

—

189

C₈

#14523968

yes

—

190

C₈

#14524008

yes

characteristic

191

C₈

#14524008

yes

characteristic

192

C₈

#14524030

yes

characteristic

193

C₈

#14524041

yes

characteristic

194

C₂²

—

195

C₂²

#16863402

yes

characteristic

196

C₂²

#16863687

yes

characteristic

197

C₂²

#16863692

yes

characteristic

198

C₄

—

199

C₄

—

536

—

200

C₄

—

536

—

201

C₄

#16863301

yes

characteristic

202

C₄

C₂×#14523968

yes

characteristic

203

C₄

#16863687

yes

characteristic

204

C₄

#16863692

yes

characteristic

205

C₂

—

206

C₂

—

207

C₂

#22561156

yes

characteristic

208

C₂

#22562381

yes

—

	subgroup	G/H
center	C₂²	#16863687
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×#364×C₆₇	C₂
socle	C₂²×C₆₇	#476
prime radical	C₂³×C₆₇	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#364	D₆₇
hypocenter	C₆₇	#13272
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

subgroup

G/H

center

C₂²

#16863687

commutator

C₂×C₈×C₆₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₂×#364×C₆₇

C₂

C₂²×C₆₇

C₂³×C₆₇

C₂²×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

N	H ≅ G/N	type
8. #16863794	C₄	semidirect
26. #14523979	Q₈	semidirect
27. #14523977	Q₈	semidirect
110. C₆₇	#13272	semidirect

H ≅ G/N

type

8. #16863794

C₄

semidirect

26. #14523979

Q₈

semidirect

27. #14523977

Q₈

semidirect

110. C₆₇

#13272

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#16863794	2
#22560088	2
#22558315	2
#22558308	2
C₂×#364×C₆₇	2
#13272	67

subgroup

index

C₂×#364×C₆₇

Subgroups — #30774388

Conjugacy classes of subgroups

Complements

Maximal subgroups